New Exact Solutions and Conservation Laws to the Fractional-Order Fokker–Planck Equations

The main purpose of this paper is to present a new approach to achieving analytical solutions of parameter containing fractional-order differential equations. Using the nonlinear self-adjoint notion, approximate solutions, conservation laws and symmetries of these equations are also obtained via a new formulation of an improved form of the Noether’s theorem. It is indicated that invariant solutions, reduced equations, perturbed or unperturbed symmetries and conservation laws can be obtained by applying a nonlinear self-adjoint notion. The method is applied to the time fractional-order Fokker–Planck equation. We obtained new results in a highly efficient and elegant manner.


Introduction
Fractional partial differential equations are a generalization of classical ordinary calculus with utilizations of integrals and derivatives with an arbitrary order. In the last decade, these equations were employed in various scientific and engineering phenomena including fluid mechanics, gas dynamics, nonlinear acoustics, biology, control theory, earthquake modeling, traffic flow models. There are several different types of fractional-order derivative and integral operators including the Riesz, Riemann-Liouville, Grünwald-Letnikov and Caputo fractional derivatives [1].
We are concerned with approximations using a small parameter of the Caputo and Riemann-Liouville type fractional derivative operators. Using this approximation, a fractional-order differential equation may be converted into an integer-order equation [2][3][4][5][6][7].
By the Lie symmetry techniques [8][9][10], we can obtain analytical solutions of many perturbed differential equations. Noether's theorem which was introduced by Emmy Noether in 1918 describing general concepts related to symmetry groups and conservation laws is a useful tool in the solutions of perturbed differential equations, see, e.g., [11][12][13]. Finding approximate symmetries of perturbed partial dofferential equations was first introduced by Fushchich, Shtelen and Baikov [14,15]. Because of the importance of perturbed systems to describe the natural phenomena, they generalized the Noether's theorem to approximated version. This generalization helps to find approximate conservation laws of a given system including the related topics [16,17]. For a system, approximate conservation laws is determined by approximate formal Lagrange and nonlinear self-adjointness for approximate equations [18]. We present conservation laws of fractional partial differential equations [19,20] with an effective method based on nonlinear self-adjointness.
The Fokker-Planck equations play an important role in fluid mechanics, control theory, astrophysics and quantum [21,22]. We are concerned with the perturbed fractional-order Fokker-Planck equation In which a, b are constants and D α t is fractional derivative of order α.
For the natural numbers, k, c, d, let u(x) := u be the function of x = (x 1 , x 2 , . . . , x n ) ∈ R n , we consider an fractional differential equation in the form of The partial derivative of u is denoted as (i 1 , . . . , i s = 1, . . . n, s = 1, . . . , k).
For the Caputo fractional derivative C a D k±ε In which Proposition 1. Let F be a continuously differentiable function with respect to a D α+k Then, for α = ε or α = 1 − ε, we can approximate Equation (7) as follows: in which c = max{d, r} for α = 1 − ε and c = max{d − 1, r} for α = ε.

Lie Group Analysis
We consider a differential operator of first order defined as in which Calculating the solutions of exact symmetries of the perturbed Equation (7) can be achieved.
are group of Lie point transformations under the group conditions by o(ε).

Classification of Group-Invariant Solution
We present the optimal system of approximate Fokker-Planck equation symmetries [23] by employing the fact that every s-dimensional subalgebra is equivalent to a unique member of the optimal system with an adjoint representation. If we know the infinitesimal adjoint action adg of a Lie algebra g on itself, we can reconstruct the adjoint representation AdG of the underlying Lie group.
It is clear that [Xi, Xj] is the usual commutator and ε is a parameter.

Optimal System and Exact Solutions
Consider the perturbed fractional-order Fokker-Planck equation In order to calculate the approximate symmetries of the perturbed fractional equation, we apply the extension of Equation (8) to Equation (16). Setting α = 1 − ε, we can write Equation (16) as We get symmetries of perturbed equation Equation (17) using the Maple software.
where the Kummer functions, KummerM(µ, ν, z) and KummerU(µ, ν, z) solve the differential equation zy + (ν − z)y − µy = 0. By the possession of infinitesimal generators (18), a number of adjoint representations are given as are the most general element. Eventually, we will obtain one-dimensional optimal system of Equation (18). The following symmetries are just a few members of optimal system of the perturbed Fokker-Planck equation

Case 1:
For the symmetry of V 1 = X 1 , corresponding characteristic equation is given as: integration of Equation (19) yields the following similarity variable and function thus we have Substituting Equations (20) and (21) into Equation (17), we can get the reduced equation: where solution of unperturbed part of reduced equation will be in the form Case 2: For V 3 = X 3 , using the corresponding characteristic equation and change of variables, we write We reduce the perturbed equation Equation (17) to a first order equation: Case 3: For V 5 = X 2 + X 3 , the reduced equation is: ) is a solution of unperturbed equation.

Case 4:
For component of one-dimensional optimal system V 4 , V 6 and V 7 , solutions of unperturbed part of Equation (17) are given in Table 1.  (17).

Approximate Conservation Laws
We consider approximate nonlinear self-adjointness for a system of perturbed PDEs, see, e.g., [24,25] for details. In the rest of this section, we present a formal Lagrange of perturbed Equation (12) and obtain conservation laws.

Basic Definitions for Constructing Conservation Laws
Let L be the formal Lagrange of Equation (12): hence, the adjoint equations of Equation (12) are defined as where v i represents all i th -order derivatives of variable v with respect to x, δ δu is the variational derivative written in terms of the total derivative operator D i : D i indicates the operator of total differentiation with respect to x i : we have L ≈ ϕ (0) P (0) + ε ϕ (1) and if it satisfies the nonlinear self adjoint condition: In which γ (0) and γ (1) are to be determined coefficients. Any approximate symmetry Equation (13) of Equation (12) leads to a conservation law where the components C i are obtained by In which

Approximate Conservation Laws for pfPE
By choosing approximate formal Lagrange where we obtain adjoint equation using Equation (23) as: It is easy to achieve an approximate formal Lagrange by placing Equation (29) and L ≈ L (0) + εL (1) , where Here, c 1 , c 2 , c 3 , c 4 , c 5 , a and b are arbitrary constants. Applying the formula Equations (26) and (27), we perform all computations to approximate conservation laws. Finally, we obtain where

Conclusions and Outlook
We presented a new approach for calculating new exact analytical solutions of parameter containing fractional-order equations. Using the nonlinear self-adjoint notion, approximate solutions, conservation laws and symmetries for these equations are obtained. Computational results indicate the strength of new method. We will apply the method to fractional-stochastic differential equations in a future work.

Conflicts of Interest:
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.